Biquotient vector bundles with no inverse

In previous work, the second author and others have found conditions on a homogeneous space G/H which imply that, up to stabilization, all vector bundles over admit Riemannian metrics of non-negative sectional curvature. One important ingredient their approach is Segal’s result that set form $$G\times _H V$$ for representation V H contains inverses within class. We show this cannot work biquotients $$G/\!\!/H$$ , where we consider _{H} . call such biquotient bundles. Specifically, in each dimension $$n\ge 4$$ except $$n=5$$ there simply connected n with bundle does not contain an inverse class addition, 6$$ $$n=7$$ are infinitely many homotopy types property no non-trivial has inverse. Lastly, every $$M^n = G/\!\!/H$$ G $$n\in \{2,3,5\}$$